Method and apparatus for phase unwrapping radar detections using optical flow

ABSTRACT

Radar systems are disclosed having phase measures limited to +/−π. An optical flow method considers the time derivative of the range with respect to phase (or velocity), and gives an indication of whether the phase is outside the measurable range by comparing the derivatives to forward and reverse wrap thresholds.

CROSS REFERENCE TO RELATED APPLICATION

The present application claims priority to U.S. Provisional PatentApplication Ser. No. 62/743,406, filed on Oct. 9, 2018, which isincorporated by reference in its entirety.

BACKGROUND

In an object detection system, there may be many objects movingindependently. This introduces many variables into the problem ofidentifying a location and velocity of a given object. Object detectionsystems using radar technology may implement modulation techniques thatuse phase of a received signal to provide information about the locationand movement of an object. As such, they are therefore limited by thephase range detectible (e.g., −π to +π) and, specifically, are subjectto wrapping when a maximum phase (e.g., +/−π) is exceeded. This leads toincorrect results in the information deduced from the phase information.This is referred to as phase wrapping, and is an artifact of the way thephase is calculated.

BRIEF DESCRIPTION OF THE DRAWINGS

The present application may be more fully appreciated in connection withthe following detailed description taken in conjunction with theaccompanying drawings, which may not be drawn to scale and in which likereference characters refer to like parts throughout, and in which:

FIG. 1 illustrates perspectives of an object detection system for avehicular application, according to various implementations of thesubject technology;

FIG. 2 illustrates position and velocity of the vehicles of FIG. 1 as afunction of range, according to various implementations of the subjecttechnology;

FIG. 3 illustrates phase unwrapping of phase measured by an objectdetection system, according to various implementations of the subjecttechnology;

FIG. 4 illustrates an object flow mapping of the movement of a vehicleto a two-dimensional (2-D) plane, according to various implementationsof the subject technology;

FIG. 5 illustrates movement of a vehicle and the corresponding opticalflow mapping for a vehicle as in FIG. 4, according to variousimplementations of the subject technology;

FIG. 6 is a flow diagram of a process for identifying objects as afunction of object flow mapping, according to various implementations ofthe subject technology;

FIGS. 7A, 7B, 7C, and 7D illustrate a mapping of the movement of avehicle as a function of object flow mapping using a process as in FIG.6, according to example embodiments of the present invention;

FIG. 8A illustrates a plot of velocity measures and thresholdsindicating phase wrapping, according to various examples;

FIG. 8B illustrates a plot of exemplary transmitted and received rampwaveforms in a frequency-modulated continuous-wave (FMCW) system with asawtooth waveform, according to various implementations of the subjecttechnology;

FIG. 9 illustrates a radar system incorporating an FMCW modulatedsignal, according to various implementations of the subject technology;and

FIG. 10 illustrates an object detection system, according to variousimplementations of the subject technology.

DETAILED DESCRIPTION

Methods and apparatuses to improve object detection in a radar systemare disclosed. There are many applications for these solutions,including those as illustrated herein below in a radar system for driverassist and autonomous operation of a vehicle. This is not meant to belimiting, but rather provided for clarity of understanding.

An object detection system in a vehicle is a moving sensor that istasked with understanding the environment within which it operates. Asillustrated in FIG. 1, a vehicle 104 may be traveling on a busymulti-lane highway 106 with vehicles 102 and 104 moving at a variety ofspeeds in multiple directions. The vehicle 104 operates withinenvironment 100 and must navigate other vehicles 102.

In object detection systems incorporating a radar modulation scheme,such as a frequency-modulated carrier-wave (FMCW) scheme, the differencebetween transmit and receive signals provides range information andvelocity. The velocity is deduced from the phase difference between thetransmit and receive signals.

FIG. 2 illustrates a mapping 110 of the system 105 where the vehicle 104has a sensor detecting objects within an angular range as indicated.There are several vehicles 102 indicated as 1, 2, 3, 4, 5, 6, 7, and 8,which may be moving in a forward or reverse direction or may bestationary. The vehicles are identified as 1, 2, 3, 4, 5, 6, 7, and 8and some of the vehicles, including 2, 3, 4, 5, 6, and 8 are mapped to aRange-Doppler Map (RDM) as V2, V3, V4, V5, V6, and V8, where the rangefrom vehicle 104 to a given vehicle is mapped with respect to thevelocity of the vehicle. As there is a relationship between phase andvelocity, the phase difference is graphed as corresponding to velocity.The velocity is a function of phase (i.e. velocity=f(phase)). In thepresent embodiments, the value of B is 180°, or π, and the value of A issome intermediary value. Understanding the velocity of a vehicle and itsrange provides information to anticipate where that vehicle is moving.

As velocity of a vehicle, such as vehicle 104, is measured by the phasedifference of received signals, there is a limit to the ability todirectly measure velocity due to the phase wrapping that occurs after+/−π. As used herein, phase difference will be generally referred to asphase, but the reader understands that this is a measure taken from areceived radar signal.

FIG. 3 illustrates the mapping 120 of vehicle velocities illustratingphase unwrapping. As is shown in the mapping 120, the detectable phaserange is −B (i.e. −π) to +B (i.e. +π). The phase of the vehicles issubject to wrapping when a maximum phase (i.e. +/−π) is exceeded.

As illustrated, the phase of vehicle V₂ measured by the object detectionsystem is D₂ greater than −π, or −B. However, a forward wrap thresholdof +π (e.g., refer to the forward wrap threshold π on FIG. 8A) isdetermined to be exceeded and, therefore, the phase is wrapped (the wrapthresholds will be discussed in detail in the description of FIG. 8A).The velocity of vehicle V₂ cannot be directly derived given thephase-wrapping, as the actual phase of vehicle V₂ is greater than +B, orπ. When the phase of vehicle V₂ is unwrapped, the phase differencebetween the actual phase of V₂ and +B is D₂ as is shown.

A similar situation occurs with vehicle V₆, where the phase of vehicleV₆ is measured to be D₆ greater than −π, or −B. Similar to vehicle V₂,for vehicle V₆ a forward wrap threshold of +π is determined to beexceeded and, as such, the phase is wrapped. When the phase of vehicleV₆ is unwrapped, the phase difference between the actual phase of V₆ and+B is D₆ as shown. The methods and apparatuses disclosed herein providesolutions to detect whether the phase is wrapped, to unwrap the phase,and to identify the correct velocity using optical flow concepts.

As illustrated in FIG. 4, movement of a car in three-dimensions (3-D)may be mapped to a two-dimensional (2-D) planar view using optical flowprocesses to indicate the movement. The vehicle 132 of system 130 ismoving in a direction indicated by vectors 134. These vectors 134 arethen mapped to plane 136 as vectors 138, where the mapping is indicatedas dashed lines 135. The optical flow method uses a 2-D plane (e.g.,plane 136) to describe the detection area of the sensor not shown). Fora radar sensor, the detection area is the area covered by the transmitand receive beams of the antenna, within which the radar has thecapability to detect an object. This does not necessarily consider theangular resolution of the radar, which is the capability to distinguishdifferent objects within the detection area. The optical flow is a 2-Dprojection of the physical movement of an object, as viewed from thesensor, to a 2-D displacement of pixels within an image plane. Pixelsrefer to small units within the image plane; the term “pixel” comes fromvideo frame processing, and is used herein to identify a small unitrepresenting a physical location, such as in an RDM.

Continuing with FIG. 4, the vehicle 132 is moving in the direction ofvectors 134 at a time instant. That instant is considered in relation toprior movement to determine an effective trajectory. These movementvectors 134 are then applied to multiple time instants, eachcorresponding to a scan of the environment.

FIG. 5 illustrates multiple time instances of measurements of phase ofthe vehicle 132. At the first time, to, the RDM identifies the vehicle132 at area 140. As indicated, the vehicle 132 is at close range to thesensor. At time t₁ the vehicle 132 has moved further away from thesensor, and has a greater range, as well as an increased velocity. Thisis indicated as area 142. Then at time t₂ the distance from the sensorto the vehicle 132 increases, thus the range and the velocity continueto increase, as indicated by area 144.

FIG. 6 illustrates a method for detecting if radar detection results areaffected by phase-wrapping and adjusting the results accordingly. Theprocess 200 considers components of successive RDMs from radar scans ofa field of view, wherein the radar scans are taken at sequential times,T(i−1), T(i), and T(i+1). In the illustrated example, the processcaptures sensor data, 202. This data is then presented to an opticalflow process 204, to create a time derivative map (refer to FIG. 8A),206, which is compared to threshold values, 208, to identifyphase-wrapped values. The process then determines if any phase-wrap isdetected, 210. If no phase-wrap is detected, 210, the process continuesto the next set of sensor data, 216; else, the process determines a wrapcoefficient, 212. The wrap coefficient provides information as to howmany times the phase has wrapped and, thus, as to how to adjust for anaccurate velocity. The process adjusts to the unwrapped value, 214. Theprocess then continues to the next sensor data. In this way, thereceived data is monitored to identify a velocity that is not within themeasurable range of the radar system.

FIGS. 7A, 7B, 7C, and 7D illustrate an example of the process 200 ofFIG. 6 in operation for a vehicle V₆, where the movement is increasingin range and velocity. The RDM 220 illustrated in FIG. 7A is not shownin complete form, but rather focuses on the coordinate area describingthe vehicle V₆ and its movement. The RDM 220 is an overlay for twopoints in time, a first coordinate location for to and a secondcoordinate location for t₁. The RDM 220 is shown in expanded view as RDMportion 222 illustrated in FIG. 7B corresponding to time to, and portion224 illustrated in FIG. 7C corresponding to time t₁. As the velocity isconstant, the vehicle V₆ continues to move in an expected way withrespect to the sensor. Where the sensor has a velocity different fromthat of the object (e.g., vehicle V₆), the distance between them changesand, thus, there is a change in the range. The plot 226 depicted in FIG.7D illustrates the change in pixel values resulting from the movement ofvehicle V₆. The expected value line 208 of plot 226 is an expecteddirect linear relationship between the time derivative of the range andthe measured velocity. The exemplary value 230, which is close to theexpected value line on plot 226, is an exemplary value representing theactual movement of vehicle V₆.

FIG. 8A illustrates a threshold scheme 300 for determining whether aphase is wrapped (i.e. for determining phase-wrap). There is an expecteddirect relationship between the time derivative of the range and themeasured velocity. The line describing this expected relationship (i.e.the expected value line in the plot of FIG. 8A) defines the midpointsbetween threshold values. The plot of FIG. 8A also includes a linerepresenting a forward wrap threshold of +π, a line representing areverse wrap threshold of −π, a line representing a forward wrapthreshold of +2π, and a line representing a reverse wrap threshold of−2π.

The maximum velocity (V_(max)) corresponds to a phase of +π. When arange change exceeds the forward wrap threshold of +π, this indicatesthat the detected object is moving faster than directly measurable withthe radar system and, thus, the phase has been wrapped in a forwarddirection. Similarly, where the time derivative of the range is belowthe reverse wrap threshold of −π, the object is moving in a reversedirection and exceeds the directly measurable capability of the radarsystem and, as such, the phase has been wrapped in a reverse direction.

For example, in FIG. 8A, the time derivative of the range for vehicle V₁is shown to be exceeding the forward wrap threshold of +2π; thisindicates that the phase has been wrapped twice in a forward direction.Also shown, the time derivative of the range for vehicle V₂ is shown tobe exceeding the reverse wrap threshold of −π; this indicates that thephase has been wrapped a single time in a reverse direction.

Heuristically, it may be understood that the methods disclosed hereincompare independent measures of velocity. The measurement of phase hashigh precision, but is susceptible to the aforementioned phase-wrappingambiguity. The measurement of the time derivative of the range issubject to high noise, but has no such ambiguity. The time derivativeinformation can therefore be used to resolve the phase-wrappingambiguity while retaining the precision of the phase measurementapproach.

FIG. 8B illustrates a plot of exemplary transmitted and received rampwaveforms in a frequency-modulated continuous-wave (FMCW) system with asawtooth waveform. In the plot of FIG. 8B, the x-axis denotes time (t),and the y-axis denotes frequency (f). An FMCW radar (e.g., refer to 400of FIG. 9) may use an FMCW sawtooth waveform to determine range andvelocity.

During operation of an FMCW radar, a chirp signal with a sawtoothwaveform is launched into the free space using a transmit antenna (e.g.,refer to 422 of FIG. 9) of the FMCW radar. A chirp signal is anFM-modulated signal of a known stable frequency whose instantaneousfrequency varies linearly over a fixed period of time (sweep time) by amodulating signal. The transmitted signal hits the target (e.g., avehicle) and reflects back to a receive antenna (e.g., refer to 432 ofFIG. 9) of the FMCW radar. From a single chirp, a Fast Fourier Transform(FFT) of an intermediate frequency (IF) signal (refer to FIG. 9), whichis the difference between the transmit and receive signals, can be usedto determine the range profile of the scanned area. The location of apeak (in frequency space) is proportional to the distance to thecorresponding target. By taking multiple such chirps, and noting how thephase of a particular peak changes between pulses, the velocity from therate of phase change can be deduced. In practice, this is determined byperforming an FFT as well.

In particular, for example, the FMCW radar emits an FMCW signal with asawtooth waveform having a period T (refer to the transmit signal ofFIG. 8B). For a simplified analysis, it is assumed that the signalreceived (refer to the receive signal of FIG. 8B) after refection fromthe target is a copy of the transmitted signal, delayed by propagationtime:τ=(2R)/c,  (Eq. 1)where R is the range of the target, and c is the speed of light.

The received signal is mixed (e.g., refer to 410 of FIG. 9) with anattenuated transmit signal. After low-pass filtering (e.g., refer to 408of FIG. 9), a low (differential) signal is obtained, referred to as avideo signal. The video signal is approximately sinusoidal, and itsfrequency f_(ω), constant in the time interval T−τ, equals the change ofthe transmitter frequency during time τ,f _(ω)=ατ,  (Eq. 2)where α=Δf/T is a modulation waveform slope, and Δf=f_(mx)−f_(min)(refer to the plot of FIG. 8B) is the maximum frequency deviation. Ascan be seen from equations (1) and (2), the measurement of the targetrange R is equivalent to the determining of the video signal frequencyduring the T−τ interval.

If a target with an initial range R₀ (at t₀=0) moves with some radialvelocity v, the delay will not be constant. Under the condition v<<c,the delay will be almost a linear function of time:τ≈(2/c)(R ₀ +vt)  (Eq. 3)

As the delay change is relatively slow, it can be noticed only in thephase change of the video signal. If the signal is analyzed in K numberof modulation periods, the Doppler frequency can be estimated from thephase changes, thus allowing for the target velocity to be computed.

Specifically, for example for the computation of the velocity, assumethat the FMCW radar transmit antenna emits a transmit signal u(t)=U cosϕ(t), where −∞<t<∞, whose frequency is:f(t)=(1/2π)(dϕ(t)/dt)=f _(min)+α(t−kT),  (Eq. 4)where kT−(T/2)<t<kT+(T/2), and (k=0, +/−1, . . . ), is a periodicalfunction of time as shown on the plot of FIG. 8B.

The received signal u₀(t)=U cos ϕ(t), where −∞<t<∞, reflected from atarget is delayed by the propagation time z. Upon mixing the receivedsignal u₀(t) with an attenuated copy of the transmit signal u(t) andlow-pass filtering, a video signal x(t)=cos ϕ_(ω)(t), where −∞<t<∞, isobtained. The video signal differential phase ϕ_(ω)(t) can be describedby equation:ϕ_(ω)(t _(k))=2π[f ₀τ₀ +kf _(d) T+(f _(ω) +f _(d))t _(k)],  (Eq. 5)where t_(k)=t−kT and (k=0, +/−1, . . . ), and −T/2+τ<t_(k)<T/2, andwhere f_(d)=(2v/c)f₀ is a Doppler (velocity) frequency of a signal, andf_(ω)=ατ₀ is a video frequency value corresponding to a target at rangeR₀. The maximum unambiguously measured velocity is equal to:V _(max) =c/(4Tf ₀)  (Eq. 6)

FIG. 9 illustrates a radar system incorporating an FMCW modulatedsignal. The system 400 includes a radar transceiver 402 coupled to aradar control unit 450 and an RDM process unit 440. The radartransceiver 402 includes a synthesizer 404 to generate a frequencymodulated signal that is transmitted at transmit (Tx) antenna 422controlled by Tx front end module 420. The signal frequency increaseslinearly with time, such as a sawtooth wave, enabling time delaycalculations to identify the range or distance to a detected object. Thesystem 400 includes a mixer 410 that receives the transmit signal fromsynthesizer 404 and the received signal from receiver (Rx) front endmodule 430 via Rx antenna 432, and then outputs a signal at anintermediate frequency (IF) (i.e. IF signal). The IF is the differencein phase of the transmit and receive signals. The IF signal is providedto low pass filter (LPF) 408 for filtering and then to ananalog-to-digital converter (ADC) 406 for conversion from an analogsignal to a digital signal. The RDM process unit 440 uses the comparisoninformation to determine range and velocity. The range is proportionalto the frequency of the IF signal, and the velocity is proportional tothe phase difference of the IF signal. The change in position of anobject is reflected in the return time and, thus, the IF phase. In thisway, both the range and velocity are calculated using the FMCW signal inthe radar system 400.

FIG. 10 illustrates an object detection system 500 having an opticalflow module 550 including a range derivative module 552, a thresholdcomparison module 554, and a phase resolution module 556. The rangederivative module 552 is operable to calculate changes in rangemeasurements as time derivatives, the threshold comparison module 554 isoperable to compare the time derivatives to wrap threshold limits (e.g.,a forward wrap threshold and a reverse wrap threshold), and the phaseresolution module 556 is operable to correct a velocity estimation if atleast one time derivative is outside the wrap threshold limits, therebycorrecting the velocity estimation.

The optical flow module 550 operates in coordination with the radarmodules, including radar capture module 504, range measurement module506, velocity estimation module 508 and RDM process unit 540. The radarcapture unit 504 receives reflected FMCW signals, and compares frequencyand phase between transmitted and received signals. Changes in frequencycorrespond to distance or range, and changes in phase correspond tovelocity of detected objects. The range measurement module 506determines frequency changes and generates range data. Velocityestimation module 508 determines phase changes and generates velocitydata. The output of modules 506 and 508 are input to RDM process unit540 to generate RDMs. The velocity estimation module 508 may measureambiguous velocities depending on the phase difference detected. If thephase difference exceeds the measurement limits of the object detectionsystem 500, then the phase will wrap around and indicate an incorrectphase. This is the case where the system interprets a phase shift of π/4as the same as a phase shift of 5π/4, 9π/4, and so forth. The opticalflow module 550 identifies these ambiguous measurements ofphase-wrapping and enables the system 500 to identify the correct phaseand, thus, a more accurate velocity.

In such radar systems, the phase measure is limited to +/−π, as anyvalue greater or less than this introduces an ambiguity into thevelocity estimation. When the phase is outside of this range, the phasewraps around and a direct application to velocity is not accurate. Toresolve this issue, the disclosure herein applies optical flowtechniques. The optical flow considers the time derivative of the rangerelative to the velocity estimated from the phase, and gives anindication of whether the phase measure is outside the measurable rangeby comparing the derivatives to forward and reverse thresholds.

The disclosed radar system (e.g., radar system 400 of FIG. 9) mayimplement the various aspects, configurations, processes and modulesdescribed throughout this description. The radar system is configuredfor placement in an autonomous driving system or in another structure inan environment (e.g., buildings, billboards along roads, road signs,traffic lights, etc.) to complement and supplement information ofindividual vehicles, devices and so forth. The radar system scans theenvironment, and may incorporate infrastructure information and data, toalert drivers and vehicles as to conditions in their path or surroundingenvironment. The radar system is also able to identify targets andactions within the environment. The various examples described hereinsupport autonomous driving with improved sensor performance,all-weather/all-condition detection, advanced decision-making algorithmsand interaction with other sensors through sensor fusion. The radarsystem leverages intelligent metamaterial antenna structures andartificial intelligence (AI) techniques to create a truly intelligentdigital eye for autonomous vehicles, which can include Level 1, Level 2,Level 3, Level 4, or Level 5 vehicles, i.e. any vehicle having somecapability of autonomous driving, from requiring some driver assistanceto full automation.

It is appreciated that the previous description of the disclosedexamples is provided to enable any person skilled in the art to make oruse the present disclosure. Various modifications to these examples willbe readily apparent to those skilled in the art, and the genericprinciples defined herein may be applied to other examples withoutdeparting from the spirit or scope of the disclosure. Thus, the presentdisclosure is not intended to be limited to the examples shown hereinbut is to be accorded the widest scope consistent with the principlesand novel features disclosed herein.

Where methods described above indicate certain events occurring incertain order, those of ordinary skill in the art having the benefit ofthis disclosure would recognize that the ordering may be modified andthat such modifications are in accordance with the variations of thepresent disclosure. Additionally, parts of methods may be performedconcurrently in a parallel process when possible, as well as performedsequentially. In addition, more steps or less steps of the methods maybe performed. Accordingly, embodiments are intended to exemplifyalternatives, modifications, and equivalents that may fall within thescope of the claims.

What is claimed is:
 1. A method for determining a velocity of an objectdetected by a radar system, the method comprising: capturing radarsensor data from a reflected signal; determining a plurality of rangemeasurements for detected objects based on frequency changes in theradar sensor data; determining a plurality of velocity estimations forthe detected objects based on phase changes in the radar sensor data;and initiating an optical flow process to confirm the plurality ofvelocity estimations, the optical flow process comprising: calculatingchanges in the plurality of range measurements as time derivatives;comparing the time derivatives to threshold limits that comprise aforward wrap threshold and a reverse wrap threshold; and correcting atleast one velocity estimation of the plurality of velocity estimationswhen at least one of the time derivatives is outside a corresponding oneof the threshold limits.
 2. The method as in claim 1, wherein one ormore of the time derivatives that falls outside the threshold limitsindicate phase-wrapped ambiguous velocity estimations.
 3. The method asin claim 2, wherein the radar system has a field of view, the methodfurther comprising dividing the field of view into a plurality ofpixels.
 4. The method as in claim 3, wherein calculating the changes inthe plurality of range measurements as the time derivatives comprisescomparing range measurements for each pixel in the field of view.
 5. Themethod as in claim 1, wherein the threshold limits identify unambiguousvelocity estimations.
 6. The method as in claim 5, wherein correctingthe at least one velocity estimation of the plurality of velocityestimations comprises unwrapping a corresponding phase of the timederivative.
 7. The method as in claim 6, further comprising: comparing aphase of a received signal to a phase of a transmittedfrequency-modulated carrier-wave (FMCW) signal; determining a phasedifference between the received signal and the transmitted FMCW signal;and identifying a velocity corresponding to the phase difference.
 8. Themethod as in claim 7, wherein the transmitted FMCW signal has a sawtoothwaveform.
 9. The method as in claim 1, wherein the radar system islocated on a vehicle.
 10. The method as in claim 9, wherein the vehicleis an autonomous vehicle.
 11. The method as in claim 1, wherein theradar system is located on a structure and the structure is one of abuilding, a billboard, a road sign, or a traffic light.
 12. The methodas in claim 1, wherein the forward warp threshold is +π or +2π, and thereverse warp threshold is −π or −2π.
 13. A system for determining avelocity of an object detected by a radar system, the system comprising:a radar capture module configured to capture radar sensor data from areflected signal; a range measurement module configured to determine aplurality of range measurements for detected objects based on frequencychanges in the radar sensor data; a velocity estimation unit configuredto determine a plurality of velocity estimations for the detectedobjects based on phase changes in the radar sensor data; and an opticalflow module configured to confirm and correct the plurality of velocityestimations by comparing the plurality of velocity estimations withthreshold limits, wherein the threshold limits comprise a forward wrapthreshold and a reverse wrap threshold.
 14. The system as in claim 13,wherein the optical flow module comprises: a range derivative moduleconfigured to calculate changes in the plurality of range measurementsas time derivatives; a threshold comparison module configured to comparethe time derivatives to the threshold limits; and a resolution moduleconfigured to correct a velocity estimation of the plurality of velocityestimations when at least one time derivative is outside the thresholdlimits, thereby correcting the velocity estimation.
 15. A radar system,comprising: a radar transceiver configured to prepare modulated transmitsignals and to receive reflections of the modulated transmit signals; arange Doppler map (RDM) processing unit coupled to the radartransceiver, the RDM processing unit adapted to capture received dataand to generate RDM information over a field of view, and an opticalflow module adapted to calculate changes in received data over time toidentify an ambiguous measurement and correct the ambiguous measurementby comparing the ambiguous measurement with threshold limits, whereinthe threshold limits comprise a forward wrap threshold and a reversewrap threshold.
 16. The radar system as in claim 15, wherein the opticalflow module is configured to calculate changes in range measurement overtime and compare the changes in range measurements to measurementcapabilities of the radar system.
 17. The radar system as in claim 15,wherein the radar system is located on a vehicle.
 18. The radar systemas in claim 15, wherein the radar system is located on a structure andthe structure is one of a building, a billboard, a road sign, or atraffic light.
 19. The radar system as in claim 15, wherein the forwardwarp threshold is +π or +2π, and the reverse warp threshold is −π or−2π.
 20. The system as in claim 13, wherein the forward warp thresholdis +π or +2π, and the reverse warp threshold is −π or −2π.